Theorem un0.1 44750

Description: ⊤ is the constant true, a tautology (see df-tru 1540). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
un0.1.1	⊢ (   ⊤   ▶   𝜑   )
un0.1.2	⊢ (   𝜓   ▶   𝜒   )
un0.1.3	⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )

Assertion

Ref	Expression
un0.1	⊢ (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1

Step	Hyp	Ref	Expression
1		un0.1.1	. . . 4 ⊢ (   ⊤   ▶   𝜑   )
2	1	in1 44542	. . 3 ⊢ (⊤ → 𝜑)
3		un0.1.2	. . . 4 ⊢ (   𝜓   ▶   𝜒   )
4	3	in1 44542	. . 3 ⊢ (𝜓 → 𝜒)
5		un0.1.3	. . . 4 ⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )
6	5	dfvd2ani 44554	. . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃)
7	2, 4, 6	uun0.1 44749	. 2 ⊢ (𝜓 → 𝜃)
8	7	dfvd1ir 44544	1 ⊢ (   𝜓   ▶   𝜃   )

Syntax hints:  ⊤wtru 1538  (   wvd1 44540  (   wvhc2 44551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-vd1 44541  df-vhc2 44552

This theorem is referenced by:  sspwimpVD  44890